60.7.41 problem 1638 (6.48)
Internal
problem
ID
[11591]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1638
(6.48)
Date
solved
:
Sunday, March 30, 2025 at 08:30:08 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right )&=0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 211
ode:=diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*sin(y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
4 \int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_1 +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} a^{2}+\int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_1 +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} -c_2 -x &= 0 \\
-4 \int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_1 +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} a^{2}-\int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_1 +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} -c_2 -x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.486 (sec). Leaf size: 338
ode=b*Sin[y[x]] + a*D[y[x],x]^2 + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} \sin (K[1])dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} \sin (K[1])dK[1]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {2 \int _1^{K[2]}-b e^{2 a K[1]} \sin (K[1])dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} \sin (K[1])dK[1]}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {2 \int _1^{K[3]}-b e^{2 a K[1]} \sin (K[1])dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} \sin (K[1])dK[1]}}dK[3]\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**2 + b*sin(y(x)) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-(b*sin(y(x)) + Derivative(y(x), (x, 2)))/a) + Derivative(y(x), x) cannot be solved by the factorable group method